Bessel interpolation formula

A formula which is defined as half the sum of the Gauss formula (cf. Gauss interpolation formula) for forward interpolation on the nodes

at the point :

(1)

and the Gauss formula of the same order for backward interpolation with respect to the node , i.e. with respect to the population of nodes

(2)

Putting

Bessel's interpolation formula assumes the form ([1], [2]):

(3)

Bessel's interpolation formula has certain advantages over Gauss' formulas (1), (2); in particular, if the interpolation is at the middle of the segment, i.e. at , all coefficients at the differences of odd orders vanish. If the last term on the right-hand side of (3) is omitted, the polynomial , which is not a proper interpolation polynomial (it coincides with only in the nodes ), represents a better estimate of the residual term (cf. Interpolation formula) than the interpolation polynomial of the same degree. Thus, for instance, if , the estimate of the last term using the polynomial which is most frequently employed

written with respect to the nodes , is almost 8 times better than that of the interpolation polynomial written with respect to the nodes or ([2]).

References

[1]	I.S. Berezin, N.P. Zhidkov, "Computing methods" , 1 , Pergamon (1973) (Translated from Russian)
[2]	N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)

References

[a1]	M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1970)
[a2]	F.B. Hildebrand, "Introduction to numerical analysis" , Addison-Wesley (1956)